Intersection (set theory) | EPFL Graph Search

In set theory, the intersection of two sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A. Notation and terminology Intersection is written using the symbol "\cap" between the terms; that is, in infix notation. For example: {1,2,3}\cap{2,3,4}={2,3} {1,2,3}\cap{4,5,6}=\varnothing \Z\cap\N=\N {x\in\R:x^2=1}\cap\N={1} The intersection of more than two sets (generalized intersection) can be written as: \bigcap_{i=1}^n A_i which is similar to capital-sigma notation. For an explanation of the symbols used in this article, refer to the table of mathematical symbols. Definition The intersection of two sets A

Calculation of geological uncertainties associated with 3-D geological models

Ian Pomian-Srzednicki

3-D geological models are built with data collected in the field such as boreholes, geophysical measurements, pilot shafts or geological mapping. Unfortunately, these data are always limited in number. It implies that geological information is sparse and subsurface models are thus always built of both subjective interpretation and mathematical interpolation/extrapolation techniques. These models are therefore uncertain and this uncertainty is rarely pointed out in a geological prognosis. Our study proposes to bring a new methodology for the evaluation of geological uncertainties related to 3-D subsurface models and to test its potential use. The methodology we propose is based on the 3-D subsurface model, which is here considered as the most probable prediction (notion of best guess). The various geological interfaces that compose the subsurface model are handled individually as Gaussian random fields. At each location of an interface, the random function Z(u) describing the position of this interface is composed of a deterministic part m(u) which represents the expected position, and a random part σ(u)ε(u) which describes fluctuations around the predicted position. Then, a model of spatial variability (a variogram function γ(h)) is proposed in order to condition the random field according to available observations. Several structural constraints, such as the shape of folds and the thickness of layers can also be accounted for in this model. At this point, we are able to estimate the local variance all over the study area by the application of the kriging technique. Finally, the variability is converted into three-dimensional information by calculating probabilities, this describes the occurrence of the various rock masses that are present in the study area. The probabilities are calculated according to intersection rules that govern the stratigraphic sequence of the subsurface model, and they allow us to build a probabilistic model of subsurface structures in the form of a three-dimensional probability field. All of this has been incorporated in a computer program.

EPFL2002

Uniform s-Cross-Intersecting Families

Andrei Kupavskii

In this paper we study a question related to the classical Erdos-Ko-Rado theorem, which states that any family of k-element subsets of the set [n] = {1,..., n} in which any two sets intersect has cardinality at most ((n-1)(k-1)). We say that two non-empty families A, B subset of (([n])(k)) are s-cross-intersecting if, for any A is an element of A, B is an element of B, we have |A boolean AND B| >= s. In this paper we determine the maximum of |A| + |B| for all n. This generalizes a result of Hilton and Milner, who determined the maximum of |A| + |B| for nonempty 1-cross-intersecting families.

Cambridge Univ Press2017

Counting Intersecting and Pairs of Cross-Intersecting Families

Andrei Kupavskii

A family of subsets of {1, ... , n} is called intersecting if any two of its sets intersect. A classical result in extremal combinatorics due to Erdos, Ko and Rado determines the maximum size of an intersecting family of k-subsets of {1, ... , n}. In this paper we study the following problem: How many intersecting families of k-subsets of {1, ... , n} are there? Improving a result of Balogh, Das, Delcourt, Liu and Sharifzadeh, we determine this quantity asymptotically for n >= 2k + 2+ 2 root k log k and k -> infinity. Moreover, under the same assumptions we also determine asymptotically the number of non-trivial intersecting families, that is, intersecting families for which the intersection of all sets is empty. We obtain analogous results for pairs of cross-intersecting families.

Cambridge Univ Press2018

Calculation of geological uncertainties associated with 3-D geological models

Ian Pomian-Srzednicki

EPFL2002